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A point P moves such that the sum of the slopes of the normals drawn from P to the hyperbola xy = 16 equals the sum of the ordinates of the feet of those normals. The locus of P is x² = k*y. Find the value of k.

1. 8
2. 16
3. 32
4. 4

Correct answer: 8

Solution

For hyperbola xy = 16, a normal at point (4t, 4/t) has slope t² (since dy/dx = -y/x = -1/t², so normal slope = t²). If P = (h,k₀) lies on the normal, it satisfies k₀ - 4/t = t²*(h - 4t). Expanding and collecting gives a degree-4 equation in t. Sum of slopes = sum of t_i² = k₀ (sum of ordinates = sum of 4/t_i). Working out the Vieta relations gives h² = 8k₀, so k = 8.

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